I-Total Coloring and VI-Total Coloring of mC 4 Vertex-Distinguished by Multiple Sets

: We give the optimal I-(VI)total colorings of mC 4 which are vertex distinguished by multiple sets by the use of the method of constructing a matrix whose entries are the suitable multiple sets or empty sets and the method of distributing color set in advance. Thereby we obtain I-(VI)total chromatic numbers of mC 4 which are vertex-distinguished by multiple sets.


Introduction
All graphs discussed in this paper are simple, nondirected graphs. Many conclusions have been obtained regarding the vertex-distinguished proper edge coloring [1][2][3] and vertex-distinguished general edge coloring [4][5][6][7] of graphs. In 2008, Zhang et al [8] proposed vertexdistinguished total coloring and related conjectures of graphs. In 2014, Chen et al [9] introduced vertexdistinguished I-total coloring and related conjectures of graphs. Many studies have been made on vertexdistinguished I-(VI-)total colorings of graphs [10][11][12] . In this study, we consider vertex-distinguished I-(VI-)total colorings of mC 4 by multiple sets.
Let G be a simple graph. Suppose a mapping f: V È E ®{12l} is a general total coloring of G (not necessarily proper). If "uv Î V, and uv are adjacent vertices, we have f (u) ¹ f (v), and if uvvw ÎEuv ¹ vw, we have f (uv) ¹ f (vw), then f is called the I-total coloring of G. If any two adjacent edges of G receives different colors, then f is called VI-total coloring of G. Obviously, I-total coloring is VI-total coloring, and the reverse is uncertain. For an I-total coloring (resp.VI-total coloring) f of G, if l colors are used, then f is called l-I-total coloring of G (resp.l-VI-total coloring). Note that when we refer to the l-I-total coloring (resp. l-VI-total coloring) of graph, we always assume that the colors used are 12l.
Let f be a general total coloring of G. For any vertex x in G, C ͂ f (x) denotes the multiple set of colors of vertex x and edges that are incident of vertex x. C ͂ f (x) is said to be the color set of x under f. No confusion arises when vertex-distinguished by multiple sets. Let χ͂ i vt (G) = min{l|G has l-I-total coloring which is vertex-distinguished by multiple sets} and χ͂ vi vt (G) = min{l|G has l-VI-total coloring which is vertex-distinguished by multiple sets}.
Then, χ͂ i vt (G) is called the I-total chromatic number of G which is vertex-distinguished by multiple sets. Similarly, χ͂ vi vt (G) is called the VI-total chromatic number of G which is vertex-distinguished by multiple sets. Let n i (G) represent the number of vertices of degree i. Suppose that Proof Obviously, I-total coloring is VI-total coloring. Thus χ͂ vi vt (G) ≤ χ͂ i vt (G).
Set t = χ͂ vi vt (G). G has t-VI-total coloring which are vertex-distinguished by multiple sets. For δ ≤ i ≤ D. Considering the vertices of the degree i, we obtain . This completes the proof.

Preliminaries
We first define a matrix A l´(l -1) , for any l ≥ 4, It is comprised by all the elements which are only in i 1 -, i 2 -,⋯, or i r -th rows but also in j 1 -, j 2 -, ⋯, or j s -th columns of A l ×(l -1) . The following six schemes are presented for the I-total coloring of C 4 which are vertexdistinguished by multiple sets. Note that all lowercase letters represent different colors. In Fig. 1(a), the color set of each vertex of C 4 is {aab}{bba}{aac}{cca}. This coloring scheme is Co1(a; b; c).
In Fig. 1(b), the color set of each vertex of C 4 is {abc}{cda}{abe}{efa}. This coloring scheme is Co2(ab; cd; ef ).
In Fig. 1(c), the color set of each vertex of C 4 is {abb}{baa}{acd}{dea}. This coloring scheme is Co3(a; b; cde).
In Fig. 2(a), the color set of each vertex of C 4 is {acb}{bda}{aeb}{bfa}. This coloring scheme is Co4(a; b; c; d; e; f ). In Fig. 2(b), the color set of each vertex of C 4 is {afb}{bcd}{dae}{eba}. This coloring scheme is Co5(a; b; cd; e; f ).
In Fig. 2(c), the color set of each vertex of C 4 is { fab}{bca}{abd}{def }. This coloring scheme is Co6(ab; c; def ). Lemma 1 When 1 ≤ j ≤ l -2 (j is an odd number), are the color sets of the vertices under I-total coloring of C 4 which are vertex-distinguished by multiple sets in Fig. 1(a).
Lemma 2 When i º 0 (mod 2) j º 1 (mod 2), and { jj + il}{ j + 1j + il}{ j + 1j + i + 1l} are the color sets of the vertices under I-total coloring of C 4 which are vertex-distinguished by multiple sets in Fig. 1(b).
group has four 3-subsets. These are the color sets of the vertices under I-total coloring of C 4 which are vertexdistinguished by multiple sets. Proof We use Lemmas 1 and 2, and only consider the remaining entries of A l´(l -1) .
and each group has four 3-subsets. These are the color sets of the vertices under I-total coloring of C 4 which are vertex-distinguished by multiple sets. Proof We use Lemmas 1 and 2, and only consider the remaining entries of A l´(l -1) .
For the remaining entries in 12l -3l -2l -1 columns, the grouping is obtained and the corresponding coloring scheme is determined using Lemma 3 Case 2 Ⅰ . For j º3 (mod 8)3 ≤ j ≤ l -8, considering the remaining entries of the jj + 1j + 2j + 3j + 4j + 5j + 6j + 7 columns, the grouping is obtained and the corresponding coloring scheme is determined using Lemma 3 Case 2 Ⅱ. For the six remaining entries in l - This leaves the 3-subsets {l -7l -2l}{l -5l -2l}. Case 2: l º 5 (mod 8).
For each j º 1 (mod 8)1 ≤ j ≤ l -5, considering the remaining entries in jj + 2j + 4j + 6 columns, the grouping is obtained and the corresponding coloring scheme is determined using Lemma 3 Case 1.
This leaves the 3- ú ú ú ú groups, and each group has four 3-subsets. These are the color sets of the vertices under I-total coloring of C 4 which are vertexdistinguished by multiple sets. Proof We use Lemmas 1 and 2, and only consider the entries of A l´(l -1) .
For each j º 1 (mod 8)1 ≤ j ≤ l -7, considering the remaining entries in columns jj + 2j + 4j + 6, the grouping is obtained and the corresponding coloring scheme is determined using Lemma 3 Case 1.
For the remaining entries of 12l -3l -2l -1 columns, the grouping is obtained and the corresponding coloring scheme is determined using Lemma 3 Case 2 Ⅰ. For j º 3 (mod 8)3 ≤ j ≤ l -6, considering the remaining entries of the jj + 1j +2j + 3j + 4j + 5j + 6j + 7 columns, the grouping is obtained and the corresponding coloring scheme is determined using Lemma 3 Case 2.
For each j º 1 (mod 8)1 ≤ j ≤ l -3, considering the remaining entries of the j j + 2 j + 4 j + 6 columns, the grouping is obtained and the corresponding coloring scheme is determined using Lemma 3 Case 1.

Proof
Obviously, there is l = ζ ͂ (mC 4 )≤ χ͂ i vt (G). Therefore, we can directly give the l-I-total coloring of mC 4 which are vertex-distinguished by multiple sets.

Conclusion
In this study, the I-(VI-)total chromatic numbers of mC 4 have been obtained, which are vertex-distinguished by multiple sets. According to the characteristics of the cycles and multiple sets, the mC n (even number) of the I-(VI-)total chromatic numbers and VI-total of the multiple sets can be similarly obtained according to the above methods. That is, if 2 ( ) l -1 2 m ≥ 1l ≥ 3 is satisfied, then χ͂ i vt (mC n ) = l and χ͂ vi vt (mC n ) = l, and two cases of recursive boundary conditions can be inferred in the proof process: if 3 ∤ n, then 2n; if 3|n, then 6n. The I-(VI-)total colorings of odd order cycles which are vertex-distinguished by multiple sets will be studied at a later stage.